Unitarity corridors to exceptional points

Abstract

Non-Hermitian quantum one-parametric $N$ by $N$ matrix Hamiltonians $H^{(N)}(\lambda )$ with real spectra are considered. Their special choice $H^{(N)}(\lambda )=J^{(N)}+\lambda V^{(N)}(\lambda )$ is studied at small $\lambda$, with a general $N^2-$parametric real-matrix perturbation $\lambda V^{(N)}(\lambda )$, and with the exceptional-point-related "unperturbed" Jordan-block Hamiltonian $J^{(N)}$. A "stability corridor" $\mathcal{S}$ of the parameters $\lambda$ is then sought guaranteeing the reality of spectrum and realizing a unitary-system-evolution access to the exceptional-point boundary of stability. The corridors are then shown $N-$dependent and "narrow", corresponding to certain specific, unitarity-compatible perturbations with "admissible" matrix elements $V_{j+k,j}^{(N)}(\lambda )=\mathcal{O}({\lambda }^{(k-1)/2})$ at subscripts $k=1,2,\dots ,N-1$ and at all $j$.

21 pp., 1 fig

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